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Semantic Facts on Kripke Frames

Johannes Marti

Abstract

This paper addresses the problem of how to represent semantic

facts in possible worlds semantics. To this aim we associate a

valuation function to every world in a Kripke frame that speci-

ﬁes the language of that world. The result is a two-dimensional

semantics for which we present a complete axiomatization in a

logical language that is based on standard modal logic. Lastly,

we sketch possible philosophical applications of the framework.

Keywords: two-dimensional modal logic, meaning change

1 Introduction

Many problems that bother philosophers of language crucially concern

the relation between semantic notions, such as for instance a sentence

expression pbeing true or pmeaning that ϕ, and modal or epistemic

notions, such as ϕbeing necessary or an agent knowing or believing

that ϕ. Widely known examples are Frege’s problem of informative

identity statements, the Twin Earth examples and semantic external-

ism, Kripke’s puzzle, the problem of radical interpretation, and the

epistemicist’s account of vague expressions. Nevertheless, there seems

to be no uniﬁed logical framework in which the interaction of semantic

and epistemic notions can be investigated.

The standard logical framework for modal or epistemic notions is

modal logic interpreted with possible worlds semantics. In modal logic

semantic questions are typically not considered. The modal formula

pis true, if pis true in all the worlds that are accessible from the

current world. It is not made explicit whether pis true at a world

because of the actual meaning of pand the non-semantic facts of the

epistemic alternatives, or because at that epistemic alternative the

meaning of the sentence expression pis such that it is true there.

2Johannes Marti

Usually it seems to be assumed that semantic facts are ﬁxed and do

not vary across worlds.

In (Williamson, 1999) the modal logic B is used to deal with a

speciﬁc semantic problem. The universal modality ϕis interpreted

as “It is deﬁnite that ϕ” and used to investigate higher-order vague-

ness. This interpretation of the modal operator implies that on the

semantic side worlds diﬀer only with respect to the semantic facts but

not with respect to the atomic facts that language is about. This

approach is too restricted to a speciﬁc semantic notion to be useful

for us.

The formal system that is treated in this paper enriches possi-

ble worlds semantics with additional structure to explicitly represents

variation in semantic fact. The truth of an atomic expression at a

world is made relative to the language in which that expression is

evaluated. To access the semantic information contained in valuation

models we add a truth operator to the language of modal logic that

was already discussed in (Stalnaker, 1978). The resulting system can

be seen as a generalization of Stalnaker’s metasemantic interpretation

of two-dimensional semantics (Stalnaker, 2001, 2004).

The structure of this paper is as follows: First, in Section 2, we

introduce the semantic structures, we call them valuation models that

we use later. In Section 3 we discuss diﬀerent notions of validity

in a well-suited modal language. In Section 4 the relation to two-

dimensional semantics is clariﬁed. In Section 5 we sketch illustrative

applications of the framework to the problems of radical interpretation

and vagueness.

2 Valuation Models

The usual semantics for epistemic modal logic is based on Kripke

frames (Blackburn, Rijke, & Venema, 2002). Kripke frames are tuples

(W, R)where Wis any set and R⊆W×Wis any relation on the set

W. The elements of Ware called worlds and the relation R⊆W×W

accessibility relation.

In standard Kripke semantics one considers Kripke models which

are tuples (W, R, V ), where (W, R)is a Kripke frame and Vavalua-

tion, that is function V:Prop → PWfor a set Prop of propositional

letters. One usually takes the valuation in a Kripke model to ﬁx the

Semantic Facts on Kripke Frames 3

w1:

Vw1:¬p

Vw2:p

w2:

Vw1:p

Vw2:¬p

Figure 1: A Valuation Model

basic non-modal facts that are true at the worlds of the model. The

atomic fact pis true at all the worlds in the set V(p). In this paper we

take a diﬀerent perspective on valuations. We think of every world

in a model as intrinsically containing information on which atomic

facts hold at that world. One can then take a valuation as assigning

meanings, that is the set of worlds where an expression is true, to

atomic sentences. On this view valuations are formal representations

of languages. Diﬀerent valuations correspond to diﬀerent ways how

language might be.

To represent semantic facts in Kripke frames, and hence semantic

knowledge for frames under the epistemic interpretation, we exploit

the fact that valuations are formal representations of languages. A

world in a Kripke frame should not only provide information about

its atomic and modal facts but it should also specify the semantic

facts that hold there. To do this we associate a diﬀerent valuation

with every world of a frame. Formally, we deﬁne a valuation model

to be a tuple (W, R, Vw)w∈Wwhere (W, R)is a Kripke frame and

Vw:Prop → PWis a valuation for every w∈W, that is called the

valuation or language of the world w. The valuation Vsrepresents the

semantic facts that hold at the world w. In the situation corresponding

to the world wan atomic expression p∈Prop has the meaning Vw(p).

The appropriate intuition to have for the languages in a valuation

model is that they are diﬀerent ways how one natural language might

be. One should not think of them as being diﬀerent natural languages

such as English or Dutch.

Figure 1 is an example of a valuation model. It contains the two

worlds w1where it is raining, and w2where the sun is shining. But

the worlds do not only diﬀer in weather but also in semantic facts

because Vw1(p) = {w2}but Vw2(p) = {w1}. In the language of s1the

sentence pmeans that the sun is shining, hence pis true at w2but

4Johannes Marti

not at w1, whereas in the language of w2the sentence pmeans that

it is raining, hence pis true at w1but not at w2.

3 The Truth Operator

We now deﬁne the formal language LTwhich is based on standard

modal logic. Aside from the modal box modality it contains an oper-

ator Tto access the diﬀerent valuations in a valuation models. The

precise syntax of the formal language LTis given by the grammar:

ϕ::= p|ϕ∧ϕ| ¬ϕ|ϕ|Tϕ.

where p∈Prop is any propositional letter

The intended semantics for the T-operator is to switch the valua-

tion under which its subformula is evaluated to the valuation of the

current world. With our intuition that valuations are languages this

motivates calling Tthe truth operator for the language of the current

world. More precisely the truth conditions of LTare as follows: A for-

mula ϕ∈ LTis satisﬁed at a world w∈Wof a model (W, R, Vw)w∈W

under a valuation V:Prop → PWif w, V |=ϕwhich is deﬁned

inductively as:

w, V |=piﬀ w∈V(p),

w, V |=ϕ∧ψiﬀ w, V |=ϕand w, V |=ψ,

w, V |=¬ϕiﬀ not w, V |=ϕ,

w, V |=ϕiﬀ v, V |=ϕfor all v∈Wwith Rwv,

w, V |= Tϕiﬀ w, Vw|=ϕ.

If ϕdoes not contain any propositional letters that are not within the

scope of some truth operator its truth value does not depend on the

language Vand we can just say that ϕis true at wif w, V |=ϕfor

any language V.

There are diﬀerent notions of validity that make sense in the con-

text of valuation models. Here we only consider validity and weak

completeness though it should be mentioned all deﬁnition and results

can be easily adapted to the more general notions of logical conse-

quence and strong completeness.

The most fundamental notion of validity results from varying the

world of evaluation and the language of evaluation independently. A

Semantic Facts on Kripke Frames 5

formula ϕ∈ LTis a general validity if w, Vl|=ϕfor all worlds wand

lin any valuation model (W, R, Vw)w∈W.

Another notion of validity is obtained by requiring that the lan-

guage under which a formula is evaluated is the language of the world

in which the formula is evaluated. We deﬁne a formula ϕ∈ LTto be

actual-language valid if w, Vw|=ϕfor all worlds win any valuation

model (W, R, Vw)w∈W.

An example of an actual-language validity that is not a general

validity is the equivalence schema Tϕ↔ϕ. It is an advantage of the

set of validities as opposed to the set of actual-language validities that

they are closed under -generalization, that is ϕis valid whenever

ϕis valid.

Two further notions of logical validity result from either ﬁxing the

world of evaluation and quantifying over all languages or to ﬁx the lan-

guage of evaluation and quantify over all worlds. We deﬁne a formula

ϕto be ﬁxed-world valid if w0,Vw|=ϕfor all worlds win the valua-

tion model of the world w0. Similarly a formula ϕto be ﬁxed-language

valid if w, Vw0|=ϕfor all worlds win the valuation model of the world

w0. The notions of ﬁxed-world and ﬁxed-language validity are inter-

esting because if we ﬁx the world to be the actual world or we ﬁx the

language to be actual English then these notions correspond roughly

to the notions of validity that stem from is called interpretational

and representational semantics in (Etchemendy, 1990). Special ﬁxed-

world validities would include certain peculiar purely modal properties

of the actual world. In the context of epistemic logic this might be for

instance that a certain agent has inconsistent beliefs. Special ﬁxed-

language validities could be certain atomic sentences that are analytic

truths such as for instance the English sentence “All bachelors are un-

married.”.

We only present an axiomatization of general validity since this

is much easier than actual language validity or even ﬁxed-world or

ﬁxed-language validity. The actual completeness proofs are rather

tedious and left out. The set of general validities in LTis a normal

modal logic in which and Tare both normal modal operators. A

complete axiomatization with respect to the class of valuation models

with arbitrary accessibility relations is given by a normal modal logic

K that is additionally closed under generalization for the T-operator

6Johannes Marti

and contains the following additional axiom schemata:

T(ϕ∧ψ)↔Tϕ∧TψTTϕ↔Tϕ

T¬ϕ↔ ¬TϕTTϕ↔Tϕ

These axioms show how the truth operator distributes over the log-

ical connectives and operators. They correspond to our implicit as-

sumption that the logical symbols have the same meaning in all the

languages of a valuation model. For completeness with respect to

frames that have a transitive, reﬂexive relation or an equivalence re-

lation one can just add the S4 respectively S5 axiom schemata for the

-modality. If one considers transitive, Euclidean frames which are

not necessarily reﬂexive but possibly serial then it is not enough to

just add the usual K45 or KD45 axiom schemata for the -modality.

One can show that in these cases the additional axiom T(ϕ→ϕ)

is needed to obtain completeness.

4 Two-Dimensional Semantics

Valuation models can be seen as a formalization of two-dimensional

semantics. The two-dimensional framework has been originally in-

troduced to give an account of the meaning for context-dependent

expressions such as indexicals and demonstratives (Kaplan, 1978).

Later, (Stalnaker, 1978) used a two-dimensional framework to model

how the meaning of an utterance can depend on the facts of the con-

text of utterance. The contextual factors that inﬂuence the meaning

of an utterance also include the possibility that the meaning of the

uttered expressions varies across contexts. With this interpretation of

two-dimensionalism, which has been called the “metasemantic” inter-

pretation, there is a close similarity between two-dimensional matrices

and valuation models, where the contexts in two-dimensional matrices

correspond to the languages in valuation models.

To explain the correspondence between valuation models and two-

dimensional matrices we consider again the example in Figure 1. The

situation in the valuation model of Figure 1 can also be represented

two-dimensionally by the following matrix:

p w1w2

Vw10 1

Vw21 0

Semantic Facts on Kripke Frames 7

In this matrix the columns correspond to worlds and the rows to the

languages of these worlds. As in two-dimensional semantics the box

operator quantiﬁes universally along the horizontal of the matrix, be-

cause the original accessibility relation in Figure 1 is a total relation.

It is clear that for every valuation model with a total accessibility

relation we can ﬁnd associated matrices for all the propositional let-

ters and whenever we have matrices for all propositional letters over

some ﬁxed set of worlds we can ﬁnd a corresponding valuation model

with total accessibility relation. This is the technical sense in which

valuation models are a generalization of two-dimensional matrices.

There is also a correspondence between the language LTand a

logical language suggested in (Stalnaker, 1978). We noticed in the

previous paragraph that the -modality has the same behavior in

both frameworks. In the matrix representation of valuation with to-

tal accessibility relation the T-operator shifts the valuation of formulas

along the vertical to the diagonal. This is exactly the semantics that

Stalnaker speciﬁes for the dagger †in (Stalnaker, 1978). Hence, the

two dimensional modal logic of and †is exactly the modal logic with

truth operators where the underlying -modality is S5. In the case

where is a S5 modality LTalso corresponds to the fragment of the

operators and in the system B of (Segerberg, 1973), which is an

early study on two-dimensional modal logic. It shall be pointed out

that this logic is diﬀerent from two-dimensional modal logics contain-

ing the modality and an actually operator @that has been studied

extensively for instance in (Gregory, 2001; Blackburn & Marx, 2002;

Stephanou, 2005). The actuality operators shifts along the horizontal,

the same dimension over which quantiﬁes, to the diagonal whereas

the truth operator shifts along the vertical.

Given the similarity with two-dimensionalism one can apply the

framework of valuation models to the philosophical problems that

two-dimensional semantics has been used for. This concerns Frege’s

problem of informative identity statements, the Twin Earth examples

that have been used to argue for externalism of semantic content and

Kripke’s puzzle about a logically ﬂawless individual that has appar-

ently inconsistent beliefs that it is aware of in diﬀerent languages. In

(Stalnaker, 2004) it is sketched how these puzzles can be resolved with

a two-dimensional semantics under the metasemantic interpretation.

For a more detailed treatment of these examples in two-dimensional

semantics, although not under the metasemantic interpretation, we

8Johannes Marti

refer to (Chalmers, 2002).

5 Applications

In this section we shortly discuss possible applications of valuation

models and of modal logic with truth operators.

5.1 Radical Interpretation

In an epistemic logic with truth operators one can study the problem

of radical interpretation, as it is discussed in (Davidson, 1984) and

more formally in (Lewis, 1974). One might take radical interpretation

to be the task of constructing a valuation model to represent the

interpreted subject’s mental state from the evidence that is available

for interpretation. In this section we sketch how this might work in

a simple single agent case where it is assumed that the accessibility

relation is transitive and Euclidean and hence the logic of is K45.

We do not discuss any associated philosophical diﬃculties.

As the basic evidence for interpretation we take the sentences that

a subject assents to, or does not assent to, under various circum-

stances. Such evidence we express with sentences of the form “Af-

ter the subject has come to believe that ϕthen she does, or does

not, assent to ψ.”. Here ϕis a sentence in the metalanguage of the

interpreter that describes some change in the environment that the

subject observes, and ψis a sentence of the subject’s own language.

To express such sentences in the formal language LTwe have to ﬁnd

formulas corresponding to the subject assenting to sentence ψand the

construction that something is true after the subject learned that ϕ.

We use the formula Tψto express formally that the subject as-

sents to the sentence ψ. So we require that the subject considers ψto

be true given what she believes about the meaning of the expressions

in ψ. The formulas of the form Tψcan be seen as a formal repre-

sentation of Davidson’s notion of the subject holding the sentence ψ

true in her own language. We take the assent to ψto express a belief

that Tψwhich in two-dimensional terminology is called a belief in the

diagonal proposition expressed by ψ. In (Stalnaker, 1978) it is argued

that an assertion of ϕnormally express the belief that ϕ, which is

called horizontal proposition in two-dimensional terminology. Only if

certain pragmatic principles are violated an assertion of ϕmight be

Semantic Facts on Kripke Frames 9

s0

s1:

Vs0:p

Vs1:¬a

Vs2:¬a

s2:

Vs0:¬p

Vs1:a

Vs2:a

Figure 2: The Subject’s Mental State

reinterpreted to express the diagonal Tϕ. In the context of radical

interpretation, however, it is justiﬁed to take all assertions to express

beliefs given by the formula Tϕinstead of just ϕbecause the sub-

ject’s own language might be very diﬀerent from the interpreters own

metalanguage and it is the subject’s own language that is of interest.

The whole construction “χholds after the subject learned that ϕ.”

can be modeled by formulas of the form [ϕ]χwhere [ϕ]is some sort

of update operator from dynamic epistemic logic (Baltag, Ditmarsch,

& Moss, 2008). Here we have not enough space to discuss how update

modalities can be made to work in a two-dimensional setting. For our

purposes it is enough to consider formulas of the shapes [ϕ]χand

[ϕ]¬χ, where χdoes not contain any -operators, to be equivalent

to (ϕ→χ)and ¬(ϕ→χ)respectively.

Now consider a simple example in which we observe that a subject

is assenting to the sentence aif it is not raining and assenting to ¬a

if it is raining. Moreover, the subject stays logically consistent so she

is not assenting to ¬aif it is not raining and not assenting to aif it is

raining. We can summarize this evidence with the following formulas

where we assume that pstands for the English metalanguage sentence

“It is raining.”:

[p]T¬a[¬p]Ta

[p]¬Ta[¬p]¬T¬a.

The possible mental states of the subject given the evidence can now

be taken to be all valuation models that satisfy the above formulas.

10 Johannes Marti

One such model is depicted in Figure 2. In this model the language

of the actual world s0is the metalanguage of the interpreter in which

the above formulas are evaluated. The propositional letters holding

at s0are not speciﬁed since the evidence for interpretation does not

constrain the atomic facts at the actual world. One might say that in

the model from Figure 2 ameans in the language of the subject that

it is not raining.

5.2 Vagueness

In the deﬁnition of valuation models of Section 2 languages are iden-

tiﬁed with valuations that assign a deﬁnite truth value, either true or

false, to every propositional letters at every world. One might wish

to loosen this requirement to accommodate for sentences that do not

clearly apply to a situation. An example of this phenomenon are sen-

tences containing vague expressions. In this Subsection I show how the

major frameworks that have been suggested to deal with the problem

of vague expressions (Williamson, 1994) lead to diﬀerent ways how

one might loosen the requirement that every sentence in a language

is either true or false at a world.

One possible adaption to the notion of a language that allows for

sentences to be neither true nor false would be to allow the values

of propositional letters under a valuation to be at a world to be any

element from a set Uof generalized truth values. Valuation would then

have the type Prop →UW, that is they are functions that assigns to

every propositional letter a function from states to to values in U.

In the case of standard classical valuations Uis just a two element

set. A more interesting example would be where Ucontains contain

three values: true, false, and undeﬁned or all real numbers in the

interval [0,1]. There are two ways how one could accommodate for

many-valued valuations on the syntactic side. One possibility would

be to use new operators with the meaning of “ phas value r” or “the

value of pis in A” where r∈Uand A⊆Uinstead of the usual

propositional letters. Alternatively one could also keep the standard

propositional letters and propagate the values Ualong the syntactic

tree to all formulas in the formal language LT. How this might be

done depends on the structure of U. In the three valued case one

might use the syntactic clauses of K3 or paraconsistent logic and use

ideas from (Fitting, 1992) for the modal operator.

Semantic Facts on Kripke Frames 11

Another way to obtain an account of semantic facts that does not

require every expression to be either true or false at a world would be

supervaluationism. In this case we identify a language of a world not

just with one valuation but with a set of valuations. Valuation models

are then structures of the form (W, R, Lw)w∈Wwhere L⊆(PW)Prop

is the set of valuation functions that is compatible with the semantic

facts holding at the world W. The satisfaction conditions of the truth

operator could then be changed to:

w, V |= Tϕiﬀ w, U |=ϕfor all V∈Lw.

This deﬁnition would give the truth operator the properties of super-

valuationistic super truth. It does for instance no longer distribute

over negations.

The valuation models from Section 2 with classic two-valued valu-

ations allow for an epistemicist’s solution to the problem of vagueness

if one lets the accessibility relation stand for epistemic uncertainty

and the modal for belief or knowledge. With this approach the

uncertainty that is involved in the use of vague expression is not at

the level of languages but is transferred to the epistemic uncertainty

that is modeled by the accessibility relation.

6 Conclusions and Future Work

In this paper we present valuation models as a straight-forward exten-

sion of Kripke models to represent semantic facts. We worked with

a simple modal language LTwhich has already been suggested in

two-dimensional semantics. This language is however not expressive

enough for our interpretation of the semantics. For instance one can

show that there is no formula in LTthat expresses that the proposi-

tional letter pmeans in the language of the current world that ϕ. It

is an aim for further work to investigate this problem and to extend

the formal language.

The relation between valuation models and other interpretations

of two-dimensional semantics should be investigated more thoroughly

from a philosophical perspective. Originally, two-dimensional seman-

tics has been a tool to capture context dependence and only under

the extreme metasemantic interpretation the context can also inﬂu-

ence the language in which an expression is uttered. But exactly this

12 Johannes Marti

extreme case has interested us in this paper. It is an interesting ques-

tion how the more standard context dependence that arises in the

semantics of indexicals ﬁts into the framework of valuation models.

In this paper we suggest applications of valuation models to the

problems of radical interpretation and vagueness but we do not discuss

them in any detail. We plan to investigate the application to radical

interpretation more thoroughly.

A further issue that is not addressed at all in this paper is how

diﬀerent conceptions of what semantic facts are inﬂuence the formal

model. For instance, if semantic facts are just about the linguistic

disposition of agents it might be appropriate to assume epistemic in-

trospection. It might also be interesting to investigate how the choice

between internalistic and externalistic conceptions of semantic content

inﬂuences the framework and how this aﬀects the contents of beliefs

that agents can have.

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Johannes Marti

ILLC, University of Amsterdam

P.O. Box 94242

1090 GE Amsterdam

e-mail: johannes.marti@gmail.com

URL: http://staff.science.uva.nl/~jfmarti/